I see many people in quick play, both teammates and opponents, use ill-timed ultimates to kill a single player. Ultimately (no pun intended), I believe, in a majority of cases, using solo ultimates to kill one player is ineffective. For instance, I was playing Mercy yesterday and a Roadhogg was trying desperately to kill me as I was flying away. He wanted to kill me so badly that he popped his ultimate while I just flew away to a distant Pharah. After his ultimate finished, we pushed the payload and got it to the next point.
Ultimate usage largely depends on the character and the situation, so there's not one rule that will tell you how to use an ultimate ability. However, I think there are some guidelines to consider when determining whether or not to use your ultimate ability:
1. Uses ultimates in conjunction with one another. Ultimates are far more damaging and strategic when used in conjunction with other ultimates. For instance, Mei's ultimate works great with D.va's. A Zarya/Phara combo is absolutely devastating, too. There are ostensibly hundreds of ultimate combos to talk about here, but the point is that ultimates become much more effective when you use them in conjunction with one another.
2. Use ultimates to counter other ultimate abilities. Reinhardt is my favorite character. As I have increased my skill with Reinhardt, I find myself using his ultimate to completely shut down another enemy team's coordinated ultimate efforts. For instance, on Nepal last night, the enemy team used High Noon and Mei's freeze ultimate at the same time to try to wipe our team at the capture point. I immediately hit Q, interrupted High Noon and stunned their entire team, charged out of Mei's freeze, and killed 4 of their players. If you suspect an enemy team might push with their ultimate abilities soon, save yours rather than killing one stray player with your ultimate.
3. Save healing ultimates to counter other ultimates. Granted, I am not a good healer by any means, but my team and I have started using Lucio's and Zenyatta's ults to counter enemy ultimate pushes and this strategy seems to work well in many cases.
4. Put your ego aside, as well as your obsession with medals, K/D ratio, etc. Before you use your ultimate ability, ask yourself: will this ultimate best help my team at this current moment? If not, save it. In organized play, this game basically comes down to which team manages their ultimate abilities the best.